Polymath A (mostly) technical weblog for Archivale.com

December 27, 2009

Government for Kids—author unknown

Filed under: Personal,Uncategorized — piolenc @ 1:24 pm

The article that follows is not my work. I wish that I could claim it, because it is the most succinct and powerful exposition of the essential evil of modern government that I have ever encountered, which is why I have included it in my ‘blog, even though my normal practice is to publish only my own work. Unfortunately, I do not know the name of the true author—this reached me through the Internet Samizdat network, and the ultimate source was either lost in the multiple forwarding, or never stated. If you know who wrote it, please leave a comment with the author’s name so that I can give proper credit.

I probably don’t need to say this, but I will anyway: I do not advocate actually practicing this teaching method on your children; the world at large is alienating enough, and they will need the most loving and rational home life possible to compensate for it.

The True Nature of Government

I think it is very important that we teach our children about the true nature of government. Now, at last, there is a way to give your children a basic civics course right in your own home! In my own experience as a father, I have discovered several simple devices that can illustrate to a child’s mind the principles on which the modern state deals with its citizens. You may find them helpful too.

For example, I used to play the simple card game WAR with my son. After a while, when he thoroughly understood that the higher ranking cards beat the lower ranking ones, I created a new game I called GOVERNMENT. In this game, I was Government, and I won every trick, regardless of who had the better card. My boy soon lost interest in my new game, but I like to think it taught him a valuable lesson for later in life.

When your child is a little older, you can teach him about our tax system in a way that is easy to grasp and will allow him to understand the benefits. Offer him, say, $10 to mow the lawn. When he has mowed it and asks to be paid, withhold $5 and explain that this is income tax. Give $1 of this to his younger brother, who has done nothing to deserve it, and tell him that this is “fair” because the younger brother ‘needs money too’. Also, explain that you need the other $4 yourself to cover the administrative costs of dividing the money and for various other things you need. Make him place his $5 in a savings account over which you have authority. Explain that if he is ever naughty, you will remove the money from the account without asking him. Also explain how you will be taking most of the interest he earns on that money, without his permission. Mention that if he tries to hide the money, this, in itself, will be evidence of wrongdoing and will result in you automatically taking the money from him.

Conduct random searches of his room in the small hours of the morning. Burst in unannounced. Go through all of his drawers and pockets. If he questions this, tell him you are acting on a tip-off from a mate of his who casually mentioned that you had both earned a bit of spare cash last week. If you find it, confiscate all of that money and also take his stereo and television. Tell him you are selling these and keeping the money to compensate you for having to make the raid. Also lock him in his room for a month as further punishment. When he cries at the injustice of this, tell him he is being “selfish” and “greedy” and only interested in looking after his own happiness. Explain that he should learn to sacrifice his own happiness for other people and that since he cannot be relied upon or trusted to do this voluntarily, you will use force to ensure he complies. Later in life he will thank you.

Make as many rules as possible. Leave the reasons for them obscure. Enforce them arbitrarily. Accuse your child of breaking rules you have never told him about and carefully explain that ignorance of your rules is not an excuse for breaking them. Keep him anxious that he may be violating commands you haven’t yet issued. Instill in him the feeling that rules are utterly irrational. This will prepare him for living under a democratic government. He is too young to understand the benefits of democracy, so explain this wonderful system as follows:

You, your wife and his brother get together and vote that your son should have all privileges removed, be caned, and confined to his room for a week. If he protests that you are violating his rights, patiently explain his error and tell him that the majority have voted for this punishment and nothing matters except the will of the majority. When your child has matured sufficiently to understand how the judicial system works, set a bedtime for him of, say, 10 p.m. and then send him to bed at 9 p.m. When he tearfully accuses you of breaking the rules, explain that you made the rules and you can interpret them in any way that seems appropriate to you, according to changing conditions.

Promise often to take him to the movies or the zoo, and then, at the appointed hour, recline in an easy chair with a newspaper and tell him you have changed your plans. When he screams, “but you promised!”, explain to him that it was a campaign promise and hence meaningless. Every now and then, without warning, slap your child. Then explain that this is self-defence. Tell him that you must be vigilant at all times to stop any potential enemy before he gets big enough to hurt you. This, too, your child will appreciate, not right at that moment, maybe, but later in life. If he finds this hard to accept, you can further illustrate the point as follows. Take him on a trip across town with you, to a strange neighbourhood. Walk into any random house you choose and start sorting out their domestic problems, using violence if that is what is required. Make sure you use overwhelming force to crush the family into submission—this avoids a protracted visit and becoming involved for long periods of time. Explain to your son that only a coward stands idly by whilst injustice is happening across town. Tell him we are all brothers and problems left to fester will eventually spill over into your neighbourhood. Use some of the $5 you took from your son as bus fare and to purchase a baseball bat.

Drink a bottle of whisky and then lecture him on the evils of smoking dope. If he points out your hypocrisy remind him that the majority of people drink and that, as already explained, the needs of the majority are the only moral standard.

Break up any meeting between him and more than three of his mates as being an ‘unlawful gathering’. If he strokes the cat without the cat giving its express permission, slap him hard for feline harassment.

Mark one designated spot in the yard where he can leave his bike. If he leaves it anywhere else, padlock it and demand $50 to release it. If he offends more than three times, confiscate the bike, sell it, and keep the money.

Install a CCTV system in your son’s bedroom and also record all his telephone conversations. If he protests, accuse him of having something to hide. Explain that only criminals seek privacy and that good, dutiful children relinquish their privacy in exchange for the advantages which protective parenthood offers. Remind him of the boy across town who was caught smoking dope in his bedroom by just such a CCTV system, and explain that this case justifies installing CCTV in all teenagers’ bedrooms.

Lie to your child constantly. Teach him that words mean nothing – or rather that the meanings of words are continually “evolving”, and may be tomorrow the opposite of what they are today.

Have a word with his teachers at school and ask them to share any merit marks your son achieves, with any ethnic minority students who did not get any merit marks. If he questions this policy, explain that long ago we abused the ancestors of these people, and so it is only fair that he shares the merits around to compensate their descendants. This is also probably a good time to tell him that his energy, talent and enthusiasm will not secure him a job if the quota of such ‘abused’ people has not yet been filled. Tell him talent stands for nothing – it is fairness and sharing which are important. Remind him that his primary duty is the happiness and welfare of people he does not know, and will never meet.

Ban cutlery from your home and make your son eat with his fingers. If he asks why, remind him of the youth who stabbed a cat to death last week with a fork. Explain that if just one cat is saved by the banning of cutlery, then this prohibition will be worthwhile. If he protests, question him closely about why he is intending to kill innocent cats, or accuse him of being a cat hater.

Issue him with a pass card which he must show before he can enter the house. Stand guard at the front door. When he comes home, politely but firmly take him into the spare room and question him about his movements. Ask him how much cash he has on his person. If in excess of $50, confiscate the lot as it exceeds the house rule for maximum cash allowed. Then search his rucksack and pockets. To keep him guessing, do the occasional strip search. If he protests, detain him for longer and make the search more thorough. If he gets really angry at this, hold him in a locked room until he misses his next outing or party.

If these methods sound harsh, I am only being cruel to be kind. I think it is important for children to understand the nature of the society in which we live.

I hope you found that amusing. I did when I wrote it, but on second reading, I feel a bit sick. It makes the point too plainly to avoid.

END

Introduction to Curve Fitting

Filed under: Engineering — piolenc @ 12:59 pm

When a body of empirical data—data derived from experiment or observation—must be used in design or analysis, it is often inconvenient to use the data in their raw form, for two reasons:

1. The data are discontinuous; that is, there isn’t a y for every possible value of x.

2. There is scatter; that is, there is more than one y for some values of x.

To make the data useful, the designer or analyst must fit an empirical equation to them. The equation will, if correctly formulated, give him a single, most probable value of y for each value of the argument x within the range of validity of the data. This can then be used in building the larger, comprehensive mathematical model that will contribute to rapid and accurate design decisions. What’s needed, then, is a technique, or collection of techniques, that allow such equations to be constructed and to verify that they are indeed the best possible fit to the data.

The discipline is called curve fitting. It is as important and useful as ever, but it is generally neglected in engineering curricula. Like dimensional analysis, it is being neglected and set aside at just the time when convenient and cheap computing power is making it easier to use and more effective.

Curve fitting proceeds in three stages: first rectification, to establish the type of equation that best fits the data set, then determination of the coefficients of that equation, then testing to verify conformity with the data and usefulness for design. (In the examples that occur in the post “Putting Numbers to Your Cooling System,” the data had already been rectified by plotting on a log-log grid, and our task was only to determine the coefficients of the equations.)

RECTIFICATION

By rectification, we mean plotting the data on special scales until we find a set of scales that plots the data as a straight line. The nature of the special scales needed to accomplish this then tells us what kind of equation we should choose to fit the data.

Linear Equation. The first step in any curve fit is plotting the data on a linear graph (x and y scales both linear). In the simplest case, this plot gives a straight line. In this case, the best fit to the data is a linear equation of the form y = mx + b, where m is the slope and b is the y-intercept of the line. This is also the easiest case for determining the coefficients, since m and b can often be read off the graph by inspection, and the least squares method can be applied directly to refining those values.

Effect of Log and Semilog Plotting

Plotting data on nonlinear scales implies a change of dependent variable. When plotting y = f(x) on a log or semi-log graph, for example, that amounts to substituting (log y) for y and taking the logarithm of f(x).

Exponential Relations y = a(10)mx

Supposing for instance that you plotted y = a(10)mx on a semilog graph; the result displayed is log y = mx + log a, a straight line with slope m and y-intercept a when y values are plotted on a logarithmic scale and the x scale remains linear. This type of plot rectifies this type of equation, so if your data were rectified by plotting on a log-linear graph you could expect an exponential relation like this to fit them. A more general form of this relation, y = a(10)mx + c, requires that the equation first be rearranged so that the constant c is on the left: y-c = a(10)mx. The new dependent variable is then log (y-c) = mx + log a, which makes the rectification process a bit more difficult, in that just plotting y values on a log scale won’t automatically rectify the data if they are shifted by a constant value c.

Do we have to generalize still further by replacing 10 with an arbitrary base number B? No, because of the property of logarithms that says that logB x = log10 x/log10 B. The two logs differ only by a constant factor 1/log10 B, which is captured in the constant m of the transformed equation, so plotting on a decimal log scale will rectify any exponential relation, provided that the additive constant c is taken care of. The only effect of a different base number is on the slope of the rectified plot.

Relations of the Form y = axm

Now imagine that our data are related as y = axm as were the heat transfer and fluid flow data in the post “Putting Numbers to Your Cooling System.” Taking the logarithms of both sides of the relation gives log y = m log x + log a. This will give us a straight line of slope m and y-intercept a when plotted on a log-log graph. If our data are rectified by plotting both x and y on logarithmic scales, then it is a relation of this type we should be looking for. As before, an additive constant complicates things; if the true relation is y = axm + c, then we have to consider (y-c) the new dependent variable and we need to find the value for c to fully rectify the data.

COMPUTING THE COEFFICIENTS

At this point, we have a linear equation in the form Y = mX + b, where X and Y represent either the original variables x and y if the data are best represented by a straight line, or transformed variables log x and/or log y or log(y-c) if the data fit a more complex relation. We can get a tentative value for m by choosing  widely separated points and computing the slope of a line connecting them, and of b by measuring the y-intercept of the line. If the data are scattered, however, this tentative set of coefficient values needs to be refined.

The “least squares” method allows us to arrive at the coefficients of a linear equation that best represent the data to which it is fitted. Theoretically, the fact that we’re operating on functions of the original variables skews the probability distribution for scattered data, but in most instances that fact won’t give us any trouble and can be ignored. The method gives us the means to determine two coefficients m and b in the transformed equation Y = mX + b; the additive constant c, if there is one, is determined during rectification. The two equations are:

ΣY – mΣX – nb = 0

Σ(XY) – mΣX2 – bΣX = 0

where n is the sum of the residuals—the differences between the Y values in the data and the corresponding values generated by our tentative fit. The Greek letter Σ (sigma) represents summations of the variables or combination of variables that follow it. To conveniently test adjusted values of m and b, we can set up a spreadsheet with the original data, their transformed values X and Y, cells for the various summations, and finally for the two equations above. Once the spreadsheet is in place we can quickly tell what changes in the coefficients move the results of the least squares equations closer to zero. To make the process even faster, the spreadsheet should incorporate an embedded chart plotting both the raw data points and the current line fit, plotted on the rectifying scales.

TESTING CORRELATION

Finally, we test the result we’ve achieved for fit and fitness—how well have we actually managed to model the data at hand? The tool we use for this purpose is the correlation coefficient I, defined as follows:

RECURSIVE CURVE FITTING

What if we have the best fit achievable with one of our standard equation forms, but the fit isn’t quite good enough? We may need an equation of the form y = F(x) + G(x), where we already have the function F(x), from our standard curve fit, which takes care of most of the fit, but still want to find another function G(x) to account for a wobble or a dip in the data that F(x) doesn’t take into account.  We proceed by tabulating the residues of F(x)—the differences between the local values of F(x) and the tabulated data; we may already have these in our least squares spreadsheet. Those residues become our new data set, and we proceed as above to fit an equation to them. That equation gives us our G(x), which we add to F(x). We then test the new, two-term fit by computing its correlation coefficient, which should be better than the original fit, and hopefully good enough for our purpose. If the original function F(x) was chosen on the basis of theoretical considerations, the form and amplitude of G(x) may reveal the presence of other, previously unsuspected phenomena at work in the system being observed or tested. It might also give insight into the nature of experimental or sensor error that is skewing the measurements.

EXAMPLE: SOLUBILITY OF NITROGEN AND CARBON DIOXIDE IN WATER

For our example dataset we chose the solubilities of some gases in water, as tabulated in the 32nd Edition of the Handbook of Chemistry and Physics (Chemical Rubber Publishing, 1950), page 1478. We’re cheating a little here, because of course there is no scatter in these data; the most probable value has already been assigned to each point by the scientists who compiled the data.

More reading:

Fogel, Charles M.: Introduction to Engineering Computations. (International Textbook Company, 1962)

Buy a photocopy from Archivale.com

Search for a used copy on Bookfinder

December 24, 2009

A Letter to the German Government about Homeschooling

Filed under: Personal — piolenc @ 2:58 pm

Madam Chancellor; Mr. President; Ladies and Gentlemen:

I hope that I may be permitted a brief self-introduction before coming to the substance of this correspondence. I am an American by birth and a Frenchman by derivation. I have worked, attended school and done military service in Europe, including two years in central Germany. I have studied the German language while in school in Switzerland, improved my knowledge at the US Defense Language Institute and used it while assigned to Frankfurt/Main. For as long as I have had opinions on this subject, I have considered myself a friend of Germany and the Germans. All this by way of asserting that I am not the kind of foreigner who ignorantly considers all Germans to be brutal goose-stepping “krauts” or “Boches.”

I mention that sad stereotype because it is receiving a good deal of support lately, and unfortunately that support comes from the recent actions of the Federal and Bavarian governments in the case of home-schooled children in general, and the much-persecuted Melissa Busekros and her family in particular. By now every home-schooling family in the world with an Internet connection is aware of this sinister affair, and the news is beginning to reach other concerned groups – fundamentalist Christians in particular – and to filter into the mainstream media. In short, there is a very narrow interval of time in which the German government can, if it acts quickly and decisively, prevent a catastrophe in its relations with the rest of the world, and with its own people as well.

The authorities mentioned earlier, acting under the power of a statute promulgated under National Socialism, have taken a girl away from her family to punish them for the sole “crime” of schooling her at home. In other words, the exercise of a fundamental human right has been treated as an offense. Worse, the responsible parties, far from recognizing and rectifying their error, have resorted to pseudoscience (again, reminiscent of Nazi-era doctrine) to justify their acts— diagnosing Miss Busekros as suffering from the hitherto-unknown syndrome of “school phobia” and forcing her to undergo psychiatric “treatment.” Further compounding their offense, they have resorted to arguments justifying their acts that tie them even more closely to the logic of tyranny: society, it seems, has an interest in the uniformity and conformity of education that overrides any right to choice held by the individual family. This argument is transparently specious – I refute it here only because responsible persons in positions of authority have asserted it, apparently in all seriousness. Society is after all a collection of individuals; if those persons, individually, have no right to choose the medium and content of their children’s education, then it is clear that the State’s education apparatus cannot reflect any collective input from those powerless parents. Instead, it is transparently and obviously the State’s interests that are embodied and asserted over all others by its education system, which is therefore better termed an indoctrination system.

Education is the single greatest force in the formation of a person. Indeed, it is unpleasant to imagine what the modern German nation would be like if National Socialism had survived long enough for all those educated under the old systems to die off. To place schooling under the absolute control of the State, allowing no alternative, is therefore the complete nullification of liberty. If we are not free to choose what we shall be, if we cannot inculcate our own values in our children, then all statutory and constitutional freedoms are of no value or effect. Any State that can control what we are has no trouble deciding what we do.

Over the years from 1945 until the fall of the USSR and the reunification of Germany, hundreds of thousands of American soldiers, myself included, stood guard side-by-side with Germans in central Europe, believing that we were defending a free republic against the threatened onslaught of a militant Socialist power. Words fail me in expressing my disappointment upon discovering that the régimes on either side of that border – superficially so different – nevertheless preserve the same false philosophical underpinnings. My disgust is shared by every American and European with whom I have discussed this matter. Please permit me to remind you that many of the young soldiers who patrolled the border in decades past are now men and women of stature and influence; they can be counted on to give voice and substance to their anger, and in a way that will harm —perhaps for a very long time—Germany’s standing in the civilized world and her relations with her allies. I urge you to consign this Nazi relic to the trash heap, and to cease your persecution of homeschooling parents and their children. It is not yet too late… but it soon will be.

Sincerely,

F. Marc de Piolenc
Master Sergeant, United States Army (Retired)

December 22, 2009

Early "Seabasing" Concepts – Still Relevant

Filed under: Aeronautics,Engineering,Floating Structures,Materials,Structures — piolenc @ 6:25 pm

Recently, thanks to the efforts of a friend in the States, a report collection that was formerly available only on 35mm microfilm has been scanned into PDF files. While entering the 400 or so reports into my catalog I came across a 1934 critique, by Charles P. Burgess of the US Navy’s Bureau of Aeronautics, of a proposal by Edward R. Armstrong for a chain of floating airstrips called “seadromes.” These were to consist of an overhead deck and a submerged ballast tank, connected by a double row of vertical cylinders. If that sounds familiar, it should – it’s more or less the standard configuration for modern Very Large Floating Structures (VLFS), including the US Navy’s proposed SeaBase platforms. That was a bit of a surprise to me, because none of the articles on VLFS or sea basing that I’ve seen has acknowledged Armstrong’s much earlier work, which began during WW1 and continued until his death in 1955.

But it gets more interesting, because Burgess’ critique and alternative are just as applicable to the modern proposals as they were to Armstrong’s. Noting that a small waterplane area is the ultimate reason for the stability under wave action of Armstrong’s seadromes, Burgess proposed a more shiplike unitary hull with an anvil-shaped cross section – swollen at the bottom to accomodate ballast, spreading at the top into a wide flight deck – giving a small and very fine waterplane area and much lower resistance to forward movement than the multiple prisms of Armstrong’s concept. In the process, he created a configuration now known by the acronym SWASH – Small Waterplane Area Single Hull – about thirty years before its time. Burgess seems to have been more conscious than Armstrong of the difficulties of deep-ocean anchorage; his concept emphasizes powered station-keeping, which is facilitated by the hydrodynamically favorable hull. Burgess also anticipates modern seabasing proposals, emphasizing the value of a shiplike configuration in getting out of harm’s way if the area starts to “heat up.” I’ve uploaded Burgess’ report to the Files area of the Nation-Builders group on Yahoogroups (file name is BA157.pdf).

A good article on Armstrong and his platform proposals:
http://www.americanheritage.com/articles/magazine/it/2001/1/2001_1_32_print.shtml

The back-issue archive at Popular Science magazine’s http://www.popsci.com also has many articles and news items about Armstrong’s work.

The main difference between Armstrong’s proposal (and Burgess’ counterproposal) and what is mooted now is the current emphasis on modularity. Both Armstrong and Burgess proposed unitary platforms, while nowadays the ability to assemble large units from small, identical components is highly prized – one VLFS concept even involves dynamic assembly and disassembly in situ to suit changing conditions! Armstrong’s configuration is implicitly modular – it consists largely of identical units repeating at equal intervals – which explains its prevalence in modern proposals. Burgess the naval architect, on the other hand, gives his SWASH a beautiful continuously-curved waterline in plan, so his hull could only be built as a single unit. It turns out, though, that minor changes would make Burgess’ configuration “modularizable,” and at the same time cheapen its construction considerably, without compromising its main advantages.

The main change is redesigning the load waterline to consist of a long parallel section, tapered abruptly and symmetrically at both ends. This allows the hull to consist of a variable number of identical “center” units capped with identical “end” units at bow and stern. The end units would have identical propulsion units built in, each capable of giving the whole shebang steerage way and not much more. You end up with the SWASH equivalent of a double-ended ferry, but with only enough installed power for station-keeping. Substituting waterjets with orientable nozzles for conventional screw propellers would allow even very large assemblies to be maneuvered without tugboats. The center units, containing no machinery, could be manufactured in very summary facilities much less well-equipped than standard shipyards. It might be advantageous to make the end units in regular shipbuilders’ yards.

Taking the whole idea one step further, the individual units could be built with double hulls, providing enough reserve flotation to allow them to float, albeit with little reserve buoyancy and with decks awash, even when fully flooded. This would allow them to be assembled into complete vessels or platforms on the water. End units would even be navigable under their own power when unmated and fully flooded – the machinery spaces, located in the ballast tank area, would be sealed and connected with the deck by a narrow trunk like the conning tower of an old-style submarine. This in turn would allow end units and center units to be assembled in separate areas, the end units, mated in pairs, being driven under their own power to where their center units awaited them. The mating operation itself could be carried out in open water, with the end units connecting, independently, with center units one by one until they had enough between them; then the two half-vessels would maneuver to join up.

When newly assembled, the new platform would look like a monitor without the gun-turret, deck flush with the water, but with the hull complete it would gradually be pumped dry inside, ready for fitting-out. It might even be possible to equip the propulsion units to serve as high volume, low pressure pumps, at least in the initial stages of pumping-out.

Materials and manufacturing technology are pretty much ad lib. – steel or aluminum, riveted or welded are feasible, but my favorite is of course ferrocement, which if properly executed can be longer-lived than any other material. Joining method for mating the sections is also up in the air. If the sections are made of steel and they were intended to remain assembled, welding would be the obvious method of choice; bolts are the obvious reversible method, but they are very expensive and would have to be fitted, in our hypothetical open-water assembly method, by divers working underwater and in very poor visibility. One technique that appeals to me is adapted from a system developed for assembling buildings from prefabricated panels in earthquake-prone areas, namely lacing the structure together with steel cables. For permanent assembly, the cables can be grouted into their channels; otherwise they can be secured with cable thimbles at their ends. Post-tensioning would then be possible, which would relieve bending loads on very long assemblies.

Armstrong’s patents:
Canada:
253,140
628,310

US:
1,378,394
1,511,153
1,892,125
2,248,051
2,399,611
2,963,868

France:
532,360
572,543

Burgess’ critique: US Navy Bureau of Aeronautics, Lighter than Air Section, Design Memorandum No. 157, February 1934, “A Proposal for a Single Hulled Seadrome,” by C. P. Burgess. Available from the Files section of the Nation-Builders group on Yahoogroups (see link above).

December 11, 2009

Putting Numbers to your Cooling System

Filed under: Engineering,Propulsion — piolenc @ 9:04 pm

[Originally published in Contact! magazine, Issue #62, pp 8 et seq.]


Radiator Installations


Putting Numbers to your Cooling System


Marc and his partner George E. Wright, Jr., formed Mass Flow in 1996 to pursue the design of ducted-fan-powered vehicles and to disseminate information concerning the design and integration of ducted propulsors. Their first book, Ducted Fan Design, Vol. 1, first appeared in 1997 and has just been extensively revised; Volume 2 is in preparation. MCM


INTRODUCTION


One of the difficulties that my co-author George Wright and I have encountered in preparing Volume 2 of our Ducted Fan Design series is in devising a practical procedure for designing an integral radiator installation. By “integral,” I mean only that the duct containing the radiator is a branch of the propulsive ducting—typically, an auxiliary inlet—allowing the propulsion fan to serve as the cooling fan as well. This is necessary (we think) to ensure cooling of a “buried” engine during runups and during long waits on the taxiway. Unfortunately, none of the more or less rule-of-thumb procedures we found in the light aircraft press would work for us, as the task we had set ourselves required a good match between radiator duct and propulsive duct flow where they met, and this required us to have some means of “putting numbers to it,” as Pazmany is fond of saying.

Not, I hasten to add, that this article will be useful only to those planning exotic airplanes. It is a fact, noted by Bruce Carmichael in his book Personal Aircraft Drag Reduction, that cooling drag in ordinary airplanes is often as much as ten times what theory says it must be; this is typical of the oversized, high drag installations that result when we really don’t know how much heat transfer we are getting and what the pressure drop across the radiator is likely to be under flight conditions or during ground runup.

We were therefore delighted when Contact! reprinted Prof. Miley’s AIAA paper,”Review of Liquid Cooled Aircraft Engine Installation Aerodynamics,” then made it available again in both volumes of Alternative Engines. This was a great service to firewall-forward innovators and reflects editor Myal’s recognition of the capital importance of efficient cooling in improving the overall efficiency of light aircraft.

The three key tools needed in designing a radiator installation are:

1. a mathematical model of heat transfer per unit frontal area as a function of air mass flow

2. a mathematical model of pressure drop as a function of air flow speed, and

3. valid data to plug into the two models, to obtain the parameters and get useful numeric results.

The first is needed to size the radiator for your engine, aircraft and flight regime. The second allows you to design the ducting and the cooling fan (if any), and to calculate the power loss (or drag increment) incurred by the cooling installation.

MODELING PRESSURE DROP

Miley’s paper seemed to offer a ready solution to the second requirement in the form of his equation #2,

w = a (σex Δp)b

where w represents the mass flow of air per unit time, σex is the ratio of the density of the air leaving the radiator to that approaching the radiator (which will typically be about 0.95 for most installations), Δp is the static pressure drop across the radiator (the pressure difference that you would measure with a manometer having one end upstream of the radiator and the other end just downstream), and a and b are constants to be determined from charted data for the radiator or core to be used. Solving for Δp in terms of mass flow gives:

Δp = (1/σex)(w/a)1/b

 

fig1.JPG


But when I attempted to apply it to data found in NACA Research Memorandum E7J013*, I ran into some trouble with the formula as published.

The first problem is fairly obvious with the equation in its original form, and that is that it is written in terms of mass flow, not mass flow per unit area. This makes it applicable to one radiator design, whereas what we need is something that characterizes a type of core and allows us to size the core to fit our needs. This problem is solved by rewriting in terms of mass flow per unit core frontal area w/A. Another benefit of this change is that, in the rearranged form of the equation, the quantity in brackets is more clearly related to the real independent variable, namely speed v, because mass flow divided by frontal area is air speed times air density (which for our purposes is practically constant), i.e. w/A = ρv. The original version was mathematically correct, but the constant a had to take care of the proportionality constants to convert mass flow to speed, as well as the parameter required to fit the data. The altered version makes the actual independent variable more obvious.The second problem also shows up when you try to use the equation in its original form, namely dimensions. As written, the quantity inside the brackets (that is raised to the power b) is dimensional, having the dimensions of pressure (F/L2). The chaos that this causes when it is raised to a fractional power is unbelievable. The dimensions can of course be made to come out right by adjusting the units of the constant a, but that gives units for a that have no clear physical connection to the problem, and are dependent on b. It makes sense to make the quantity inside the brackets non-dimensional. The easiest way is to substitute for Δp a ratio of pressures, Δp divided by some suitable pressure related to the problem. The quantity being normalized is a static pressure drop, so the logical normalizing quantity is the static pressure of the incoming air. With the quantity in brackets now non-dimensional, the dimensions of the constant a are M/L2T, or mass flow per unit area. Perfect! Our revised formula is then

w/A = a(σex Δp/p)b

where p is the static pressure of the air stream coming into the radiator, and A is the frontal area of the radiator core.

Rearranging this to solve for Δp gives:

Δp/p = (1/σex)(w/aA)1/b = (1/σex)(ρv/a)1/b

 

Incidentally, for the information of our non-mathematically-inclined readers, it’s worth mentioning at this point that it is very easy to determine whether a given dataset is a good fit to an equation of the form y = axb, provided that the data are plotted on a log-log chart, that is one with logarithmic scales for both the x and y axes. Such datasets give straight, or nearly straight lines on a log-log graph, and the slope of the line is the value of the exponent b! Finding the value of the exponent is as easy as selecting two extreme points (xmin, ymin) and (xmax, ymax) on the line and using the formula:

b = (log ymax – log ymin)/(log xmax – log xmin)

Another interesting property of log-log charts is that even if the axes are rescaled (values multiplied by a constant), the slope remains the same.

Once the slope is determined, only one data point (which can be one of those used to determine the slope) is needed to determine the remaining parameter.

Turning now to NACA Research Memorandum E7J01, we find there some very useful data provided by the Harrison Radiator Division of GMC in connection with the development of condensers for nuclear-powered aircraft with Rankine (steam) cycle power plants. Oh well—a core is a core! Using the slope formula above and the second form of our revised equation to fit the pressure drop chart there (Figure 2), we get an exponent 1/b of 1.79—essentially identical to the 1.8 that Hoerner4 says to expect in this kind of data.

fig2.JPG

For obtaining the remaining parameter we need to rearrange our equation to suit the way that the Harrison data are presented, namely as σΔp vs. mass flow. We assume that their σ and our σex are the same; they aren’t, but both will be close to 1 in our application, so the different definitions don’t matter much. We end up with

σΔp = p(w/aA)1/b

Solving for the parameter a gives

a = (σΔp/p)-b (w/A)

 

and plugging in the values σΔp = 30 inches of water, w/a = 600 pounds per minute per square foot from curve “A” of Figure 2, and assuming that the static pressure of the incoming air is sea-level pressure (p = 407.2 inches of water absolute) just for the sake of getting a numerical answer, we get 2555 pounds per minute per square foot. We can now generate numbers for pressure drop for this type of core for any core size for any flow rate within reason.


HEAT TRANSFER MODELING

Okay, now that we have one of our tools, it’s time to start developing the other—the heat transfer model. Looking now at heat transfer data from the same very useful NACA RM (Figure 3), we make a very pleasant discovery—the heat transfer coefficient-vs-air flow data are also a good fit to the same type of equation! The slope is .84, which is pretty close to the figure of .80 given by Hoerner as typical. Tentatively, we write our curve fit in the form:

K = k(w/A)r = k(ρv)r

where K is the heat transfer per unit time, per unit frontal area, per unit of temperature difference between the coolant and the air; k is a parameter to be determined from the data, and r is the slope that we’ve already found, namely 0.84.

Again, we want the quantity that is raised to a fractional power to be nondimensional, so we need some suitable normalizing factor. The important quantity in the brackets really is mass flow (per unit area) this time, early radiator analysis having shown that “heat transfer is a function of mass flow of air, regardless of density1” so we need a “reference” density and speed for normalization of the quantity in parentheses. The logical choice of normalizing speed is the speed of sound, and for density the density of the incoming air. The quantity inside the bracket then becomes the Mach number of the air approaching the core, multiplied by our old friend σex.

We now have:

K = k(σex v/c)r = k Mr

Where c = (1.4 p/ρ)1/2 ,the speed of sound at the pressure and density of the air approaching the radiator.

“What happened to air density ratio σex inside the brackets?” you ask. To be absolutely honest, we cheated a little. Because it is essentially constant and very close to 1, we moved it outside the brackets and incorporated it into the parameter k . We will simply divide the mass flow numbers by ρc before determining the value of k. At sea level standard conditions, c = 67,200 feet per minute, while ρ = .07382 pounds per cubic foot. As long as we’re rescaling, we divide the K values on the vertical scale by 100, to put them on the basis of heat transfer per degree Fahrenheit, rather than per 100 °F temperature difference between coolant and air.

We’re now ready to find the parameter k. Taking the point on the curve where mass flow is 500 lb/min and heat transfer coefficient K is 80 BTU/min/°F/ft2, we divide 500 by ρc to get a Mach number of .1008. Solving for k gives

k = K/Mr = 80/.1008.84 = 550 BTU/min/°F/ft2

fig3.JPG

REPLOTTING DATA

But what if the data you get are not plotted in log-log form? If the data are in the form of a table of numbers, great! If the table is a clean copy, you may be able to scan the numbers, OCR them and import them into a spreadsheet program, which will oblige you with a plot scaled any way you like. If the data are presented graphically to linear scale, then you will have to pick off data points as accurately as you can, then enter them in a spreadsheet and get your log-log plot that way.

GETTING DATA?

Now that you, the long-suffering reader, have plowed through the above, you are no doubt wondering how to use this information. Obviously, the specific numbers that we got for a, b, k and r are not useful in themselves, unless you are planning to order a half-century-old core type from Harrison (might be interesting to get their reaction to that inquiry!). What is useful is the procedure for testing data for conformity to Miley’s formula and its ilk, and for calculating the parameters that apply to each particular core that you are interested in. What you still need, in order to apply this procedure to your project, is data on the core or cores that you are thinking of using. Logically, you would expect to be able to obtain these data from the manufacturer of the core, but I honestly don’t know whether pressure drop has even been measured for any automobile radiator. Judging from the design of the auto radiator cores that I’ve seen, the manufacturers are primarily interested in ensuring adequate heat transfer under adverse conditions, with aerodynamic efficiency either completely neglected or given only secondary consideration. Mind you, all the cores I’ve seen come from older vehicles; perhaps with the trend toward smaller engines, lighter vehicles and higher efficiency in general somebody has been gathering the data we need. Maybe. Maybe we can even get it. Maybe.

If we can’t get the data, then we have to develop it ourselves (or limit ourselves to the use of heat exchangers for which the necessary data can be obtained, which might be expensive). How would we go about doing that? When I first thought about this problem some years ago I sketched out an elaborate installation that would use an immersion heater to heat water in a reservoir, a pump to circulate it though the test radiator, a kind of wind-tunnel arrangement to blow air through the radiator, and instruments to measure temperature, pressure drop and flow rates and provide a basis for controlling all this. Needless to say, the rig did not get built. Early NACA radiator test rigs anticipated my thinking by many decades, and actually made mine look simple. Later work by NACA2, however, revealed that you don’t need to have heat transfer taking place to get realistic numbers for pressure drop in service—there’s a procedure for correcting test results obtained without heat transfer so as to to take heat transfer into account. That simplifies the test rig enormously, provided of course that we don’t need to take heat transfer data (not so unrealistic, considering that in the air conditioning industry, for example, heat transfer data are often provided, while pressure drop data often are not). Also please note that if the dataset fits the model we’ve been using, we only need two reliable data points to get the two parameters. That means that the calibration of our test rig need only be accurate at two points, again making the job easier. Other, intermediate data points would have to be taken to verify conformity with the model—otherwise we would be guilty of the sin of assuming our conclusions—but “record” accuracy would only need to be achieved at two widely separated points. Worth looking into, I think.

As a last resort we might consider yet another alternative, that of reverse engineering. This would consist of taking the heat exchanger design methods of e.g. Kays and London5 and inverting them to derive heat-transfer and pressure drop characteristics from the shape and dimensions of the radiator’s core. This could get tricky, but used in conjunction with cooling systems analysis technique presented in Küchemann and Weber6 it might be doable.
FMP

REFERENCES & FURTHER READING

1. Results of Tests on Radiators for Aircraft Engines;
Part I.—Heat Dissipation and Other Properties of Radiators by H. C. Dickinson, W.B. James, and H.V. Kleinschmidt
Part II.—Water Flow through Radiator Cores by W.S James. NACA Report No. 63

2. Becker, John V. and Donald D. Baals: Simple Curves for Determining the Effects of Compressibility on Pressure Drop Through Radiators. NACA Advance Confidential Report L4I23, September 1944.

3. Humble, LeRoy V. and Ronald B. Doyle: Calculated Condenser Performance for a Steam-Turbine Power Plant for Aircraft. NACA Research Memorandum E7J01, May 20, 1948.

4. Hoerner, S.F.: Fluid-Dynamic Drag (published by the Author, 1967)

5. Kays, W.M. & A.L. London: Compact Heat Exchangers (McGraw-Hill, 1964)

6. Küchemann, Dietrich & Johanna Weber: Aerodynamics of Propulsion. (McGraw-Hill,1953)

December 7, 2009

Dimensional Analysis – a Neglected Engineering Tool

Filed under: Engineering — piolenc @ 9:41 pm

Excerpt from Ducted Fan Design, Volume 3 (Engine Cooling Systems)
by F. Marc de Piolenc & George E. Wright, Jr.
(in preparation)

Dimensional Analysis


Engineering is often ahead of science.

Now, before we are summarily executed by a lynch mob of enraged physicists, let us clarify that statement. What we mean is that engineers, in solving technical problems, must often venture into areas where science has not yet penetrated. This happens less often today than, say, in the mid-18th century, but it happens often enough to make it highly desirable that engineers learn to make the most of empirical data where no firm theoretical ground exists, or where such theory as does exist is too unwieldly to apply to technical problems.

In this volume, for example, we venture into the design of ejectors. Much empirical information exists for these contraptions. Much theoretical work has also been done, yet there is nothing that we would call a theory of ejectors in a form usable for design. Instead, much ink and many electrons have flowed constructing elaborate numerical simulation schemes which require a great deal of computer capacity and must be repeated ab initio if the slightest change is made in the device in question. An engineer who is given the task of designing an ejector with a certain specification still needs the same tools for exploiting empirical data that his forebears had available to them in the 1920’s. These are dimensional analysis and curve fitting.

Unfortunately, in their eagerness to cram as much as possible into an already overcrowded curriculum, engineering schools seem to be neglecting these very useful techniques. Dimensional analysis becomes a tedious exercise in reinventing the wheel, and curve fitting is mentioned in passing during a required lab course that everybody involved (including the instructor) wishes were not being taught. This is unfortunate, because these techniques began to be neglected at just the time when their usefulness was vastly increased by the general availability of cheap digital computers.

We have therefore made it our business to present our rough and ready versions of both these techniques, both for the edification of our readers and—frankly—to justify and explain some of our own work in preparing this book.

Dimensional Analysis


The exercise of dimensional analysis is intended as a remedy for typical problems posed by a body of empirical data resulting from observation or experiment:


  • There are too many variables, and relations between those variables are so numerous as to be intractable; in other words, the problem has too many degrees of freedom.

  • Variables measured are dimensional quantities; numerical values are valid only in a particular system of units, and only at the scale of the original experiments.

Dimensional analysis aims to collapse the large number of dependent and independent, dimensional variables into a smaller set of non-dimensional parameters, with a much smaller number of relations between them.

The new, non-dimensional quantities are formed by taking products of powers of the original, usually dimensional variables—products so constituted that the dimensions of the quantities making up the product exactly cancel. Such a product is often called a Pi (pronounced “pie”), after the Greek upper-case letter Π which is used in mathematical notation to symbolize a product, just as the capital letter sigma (Σ) is often used to represent a sum. Buckingham’s “Pi Theorem,” which formalized dimensional analysis,* basically allows you to determine, from the variables and their dimensions, how many Π products are possible, and provides rules for forming them and using them.

Where confusion sets in is that most examples of the use of the Pi Theorem presented in textbooks introduce some a priori theoretical knowledge (over and above the initial choice of relevant quantities) to guide the choice of which dimensionless products are useful, and which are not. This leaves the student scratching his head and wondering what the exercise is good for, if it requires prior knowledge of what the student is attempting to discover! Fortunately, you don’t need a complete theoretical framework to make the Pi Theorem useful. Having formed all possible Πs (and you know you’re finished when the number of Πs equals the number calculated from Buckingham’s formula), you simply apply them to the data that you have. The nondimensional products that collapse the data into a single smooth curve or family of curves are the ones to be retained. In the bad old days before desktop computers and spreadsheet programs, this testing of Π products against the available data could be very laborious, hence the textbook shortcuts that have confused generations of students.


System Quantities


We do need to know something about the system in order to use dimensional analysis effectively; that is, we need to start with a list of quantities that we think determine or characterize the system’s behavior. These quantities fall into two categories:


  • Independent parameters, that is parameters which are under the designer’s direct control. One example of an independent parameter is the tensile strength of the material from which a machine is constructed.

  • Dependent parameters, that is parameters whose values the designer controls only indirectly, through choice of the independent parameters which he thinks influence the dependent ones. One example might be the weight of a finished piece of machinery.

Usually, the dependent parameters are already chosen—they are the parameters which the analyst wants to learn to control through his choice of the independent parameters. In the examples used above, the analyst has been set the task of minimizing the weight of a certain machine, and he is seeking insight into the rules that govern that weight. Starting with the dependent parameter—weight—he will then compile a catalog of the independent parameters that he thinks, together, govern that weight. Those independent parameters might be the tensile strength of the material, its density, some parameter or combination of parameters governing the peak forces to which the machine members are subjected, and perhaps a spatial dimension or dimensions characterizing the size of the machine.

If the initial catalog of system parameters is incomplete—that is, at least one influential quantity is omitted—the results of the dimensional analysis will fail to completely characterize the system. It may even be impossible to get any correlation when the nondimensional parameters found are applied to the available data. If the chosen list of parameters is redundant—that is, if one or more of the quantities in it can be derived from other quantities listed—more nondimensional products will be found than are needed to describe the system. Both errors result in extra labor, the first by requiring the analyst to start over, the second by wasting his time applying superfluous parameters to a (possibly large) body of data. In the old days, this could be very discouraging because of the lengthy, tedious and repetitive labor involved. Today, with digital computers readily available, the penalty is much less severe and it is not unreasonable to contemplate several iterations of a dimensional analysis, each unsuccessful attempt to fit nondimensional parameters to the data refining the analyst’s insight into the system.

Dimensions and Units


A dimension is the essence of a quantity being measured, and is independent of the system of units in which it is measured. Thus units can be said to have dimensions, but dimensions don’t have units. For example, the quantities “33 meters” and “12 feet” both have the dimension of length, symbolized with L. The quantity “18 feet/second” has the dimensions of length per unit time, represented by LT-1. In the system of fundamental dimensions that we have implicitly chosen—mass, length, time or MLT—”12 feet” is a primary quantity, because it only requires the first power of a single fundamental dimension to describe it. The quantity “18 feet/second” is called secondary because it requires more than one fundamental dimension to describe it. The quantity “two square feet” is also secondary, requiring the second power of the fundamental dimension L to describe it.

It is worth noting here that there is nothing inherently fundamental about any particular system of dimensions; any system that does not contain redundancies is equally usable. In addition to the M, L, T system, engineering work frequently uses the system F (force), L, T. In fact, the US customary system of units implies an F, L, T system of fundamental dimensions, because the pound is a unit of force, while the metric system (S.I.) implies an M, L, T scheme, as the kilogram is a unit of mass. In the F, L, T system, mass is a secondary quantity with the dimensions of force divided by acceleration, or FT2L-1, while in the M, L, T system it is primary, with dimension M.

Special Dimensions and Units


Some quantities are inherently non-dimensional. As noted in the advance-ratio example below, an event, such as the completion of a cycle of alternating current or the full rotation of a shaft, has no dimension of its own. Frequency (events per unit time) thus has the dimension of inverse time, T-1.

The ways that angles are measured offer special problems in comprehension. Angles are ratios of arc length to radius, L/L; thus they are inherently nondimensional. Note, however, that when an arc is arbitrarily subdivided into degrees, points, grads or whatever, a conversion factor is needed to obtain the correct numerical value of the ratio. In other words, although angles are non-dimensional, use of arbitrary divisions to describe them still results in different numerical values for each system of divisions, just as if they were dimensional! Further confusion is caused by the assignment of a unit name to the pure arc-length/radius ratio, which is said to be expressed in radians (full circle equals 2π radians). This is a “unit” with no dimensions, used only to tell the reader that the angle is expressed as a pure ratio, and thus requires no conversion factor to allow its use in a non-dimensional parameter.

Temperature is another special case—one that purists and practical engineers love to fight about. In physical terms, temperature represents energy per unit mass—the average kinetic energy of the molecules of an ideal gas, the vibrational energy of atoms in a crystal lattice, and so on. But as actually measured, temperature is a potential, so the numerical value of temperature does not correspond directly to the energy of a substance, and temperature is measured in arbitrary systems of units because it is impractical to use a nondimensional figure. What is more, as a practical matter temperature cannot be allowed to cancel any dimensional quantity other than temperature in calculations. For all these reasons, temperature is usually treated as a separate dimension, denoted by the Greek letter theta (Θ).


Non-Dimensional Groups


To construct a Π, also called a non-dimensional group or non-dimensional parameter, we need to select combinations of quantities whose dimensions cancel. The simplest cancellation is a ratio of quantities with identical dimensions: area over area, pressure over pressure, etc. Next easiest is quantities which do not have to be raised to a power to cancel each other. One example of this kind is the “advance ratio” used in propeller work, which is speed of advance divided by the product of shaft rotation rate and propeller diameter, v/nD. In our M, L, T system of dimensions, speed v has the dimensions LT-1, n is events (rotations) per unit time or T-1 and D has the dimension of length, L. It is easy to see that the numerator and denominator have identical dimensions, and thus the quotient is nondimensional. In more complicated cases, it will be necessary to raise quantities to higher powers to get cancellation.

Universality of Non-Dimensional Groups


Note, however, that regardless of what system of fundamental dimensions we choose, a non-dimensional parameter constructed with one system of dimensions remains non-dimensional in any other; the fundamental dimensions will always cancel. Thus nondimensional parameters are not only independent of the system of units, but also of the fundamental dimensions that are chosen. Products and quotients of nondimensional groups are also nondimensional and nondimensional parameters raised to any power remain nondimensional.

Similarity and Scaling Laws


There is an important advantage to dimensional analysis that we have not yet mentioned, namely similarity and the derivation of scaling laws. If the nondimensional parameters found are truly representative of the system, then any system whose independent nondimensional parameters match those of the system which has been tested or observed will also have the same values of the dependent nondimensional parameter. This allows test results to be applied to entire families of similar systems—that is systems having the same nondimensional parameter values—even though their physical quantities may be very different. For the sake of illustration, let us suppose that the efficiency of propellers (a dependent, nondimensional parameter) was shown to depend only on their respective advance ratios (the independent nondimensional parameter). Then tests performed on a small diameter, fast turning propeller could be applied to a large, slow turning rotor if their advance ratios were equal. Thus one set of tests would apply to two pieces of machinery which were very different in both appearance and application. Considering what wind tunnel time costs, this is a strong selling point.

While the propeller example is absurdly oversimplified, entire families of hydraulic turbines have been designed with the same nondimensional specific speed, and their characteristics predicted with acceptable accuracy from tests of one member of the family.

How Many Πs?


Having established what a dimension is and gained an understanding of the notion of a nondimensional parameter or Pi, we can present Buckingham’s rule for determining the number of nondimensional parameters that can be constructed. That number is

n = m – k.

where n is the number of Πs to be formed and m is the number of quantities required to describe the system under analysis.

In the original formulation of the Pi Theorem, k was simply the number of fundamental dimensions required to form all the n quantities. A later refinement showed that k was actually the largest number of system quantities which could not form a Π between them. This number was usually equal to the number of fundamental dimensions r, but not always. It was shown, however, that if it differed from r it would be smaller, hence k ≤ r. As we will see anon, this need not cause serious trouble or delay.


A Concrete Example: Specific Weight of an Internal-Combustion Engine


From this point on, the method is best described by reference to a concrete example. In the late 80’s, Marc started on an analysis of the factors influencing the specific power (power per unit weight) of internal-combustion piston engines. That work was not completed because he found some usable empirical formulas worked out by others, but the dimensional analysis is still on file, and the topic is likely to interest readers of this book.

For reasons now forgotten, Marc chose to analyze weight and power separately, deriving Πs for each of those quantities separately before attacking the ratio of weight to power. Collecting all the variables thought to contribute to weight, power or both gave a table like the one that follows:


Dimensional Analysis of the Specific Weight of a Positive Displacement
Internal Combustion Engine


Symbol

Name of Quantity

Dimensions

M

engine mass

M

P

engine power

ML2T-3

b

cylinder bore

L

s

piston stroke

L

ω

engine operating frequency (“rpm”)

T-1

N

number of cylinders

dimensionless

ρ

density of engine material

ML-3

σ

allowable stress in engine material

ML-1T-2

p

peak pressure in cylinder

ML-1T-2

MEP

mean effective pressure of engine operating cycle

ML-1T-2


Engine Mass Analysis



The quantities thought to influence engine mass M are bore b, stroke s, number of cylinders N, the density ρ of the material from which the engine is made, its allowable stress σ and the peak pressure p occurring in the cylinders. Thus for engine mass analysis, m = 7.

The number of dimensions needed to form all these quantities, r, is 3 (M, L and T). We need to check that k, the largest number of quantities that cannot form a Π between them, is equal to r. To prove that, we only have to find one combination of three quantities that will not form a Π, because we already know that k cannot be larger than r. A quick inspection shows that b, s and M will not form a Π between them—neither b nor s has the dimension M, and engine mass M does not include the dimension L, so dimensional cancellation is impossible in this combination. There may well be other three-way combinations that won’t form a Π, but we only need one to satisfy ourselves that k = r = 3. The number of Πs to be formed is therefore:

n = m – k = 7 -3 = 4

We are looking for four nondimensional parameters—no more, no fewer.

To find the Πs we proceed as follows:

1.    First, we select a number of quantities equal to the value of k—three in this case—making sure that they are dimensionally independent (that is, they all have different dimensions). The quantities chosen should also include, between them, all the dimensions present. Marc selected b [L], p [ML-1T-2] and M [M].

2.    Products of various powers of these core quantities are then combined with the remaining dimensional variables, one at a time. In this instance, the product bα ρβ Mγ is combined in turn with each of the remaining variables s, N, ρ and σ. For each combination, values of the exponents α, β and γ that give a nondimensional product must be found.


A convenient way to start the process is to set up a table as shown below for each P to be found. In the first example, the combination of the core variables with the stroke s is shown.


Table for finding Π1

Quantities

bα

ρβ

Mγ

s

Dimensions

[L]

[ML-1T-2]

[M]

[L]

M

0

β

γ

0

L

α

β

0

1

T

0

-2β

0

0


At each intersection of a column representing a quantity and a row representing a dimension is the value of the exponent of that dimension in that quantity. For example, the power of the dimension T in the variable p is -2. When the variable as a whole is raised to the power β, the resulting exponent for dimension T is -2β, so that is entered where column p and row T intersect.


The resulting parameter Π1 must be nondimensional; put another way its dimensions must be M0L0T0. Therefore the sums of the exponents of each of the dimensions must be zero. This allows us to set up three simultaneous equations as follows, pulling the various values from the table:

M: β + γ = 0

L: α – β + 1 = 0

T: -2β = 0

The equation for the dimension T gives us β = 0 by inspection. Plugging that value into the equation for L gives α + 1 = 0, or α = -1. Since we already know that β = 0, the equation for M gives γ = 0 by inspection. Therefore:

Π1 = b-1s = s/b, the ratio of stroke to bore.

Now it happens that piston-engine aficionados refer to the “squareness” of an engine, which is essentially the ratio of bore to stroke. An engine that has a bore larger than its stroke is called “oversquare,” one with with a stroke greater than its bore “undersquare.” It would therefore be more consistent with current trade usage to make Π1 = b/s rather than s/b. Can we do this? Yes. Recalling that raising a nondimensional parameter to any power leaves it nondimensional, and that the arithmetic inverse results from raising the quantity to the power -1, we conclude that we can substitute Π1= b/s for the expression we originally derived without violating the rules of dimensional analysis.

Having shown our work in detail for the first nondimensional parameter, we will present only the tables and results for the remaining three.


Table for finding Π2

Quantities

bα

pβ

Mγ

N

Dimensions

[L]

[ML-1T-2]

[M]

[nondimensional]

M

0

β

γ

0

L

α

β

0

0

T

0

-2β

0

0

This gives the result that α, β and γ all equal zero, so the core variables disappear and the second nondimensional parameter consists only of the number of cylinders, which is nondimensional in its own right. Thus, Π2 = N.

Actually, we could have predicted this without bothering with computation, because the three core variables do not form a Π between them, and as the variable N contributes no dimensions of its own there is no nonzero power of the core variables that will make the combination nondimensional. Thus if our core variables are chosen to be a non-Π-forming group, any nondimensional quantity that they are combined with will end up forming a Π all by itself.


Table for finding Π3

Quantities

bα

pβ

Mγ

ρ

Dimensions

[L]

[ML-1T-2]

[M]

[ML-3]

M

0

β

γ

1

L

a

β

0

-3

T

0

-2β

0

0


This gives the results α = 3, β = 0, γ = -1. Therefore:

Π3 = ρb3/M.

Here, because engine mass is the quantity that we are studying, M should be in the numerator. We play the same trick as with Π1 and invert the parameter.
Now Π3 = M/ρb3.


Table for finding Π4

Quantities

bα

pβ

Mγ

σ

Dimensions

[L]

[ML-1T-2]

[M]

[ML-1T-2]

M

0

β

γ

1

L

α

β

0

-1

T

0

-2β

0

-2


We get α = 0, β = -1, γ = 0. Therefore:

Π4= σ/p


Note that this is not the only possible set of Πs. Marc could for instance have substituted s for b in the core variables, in which case Π3 would have been M/ρs3.


Engine Power Analysis



Quantities thought to contribute to engine power output P are bore b, stroke s, operating frequency ω, number of cylinders N and mean effective pressure MEP; thus m = 6. As with engine weight, the number of dimensions r required to form all these dimensional quantities is three.

As before, we need to check that k equals r. At least one combination of
three quantities—s, ω and MEP—cannot form a Π between them because the dimension M in MEP cannot be cancelled by either of the remaining two quantities, so this is confirmed.

The number of Πs to be found is therefore n = m – k = 6 – 3 = 3.

As the three quantities already chosen to verify k = r are dimensionally independent, and contain between them all the dimensions present in the problem, they are selected as the core variables, to be combined with the three remaining quantities b, N and P.


Table for finding Π5

Quantities

sα

MEPβ

ωγ

b

Dimensions

[L]

[ML-1T-2]

[T-1]

[L]

M

0

β

0

0

L

α

β

0
1

T

0

-2β

γ

0

This gives α = -1, β = 0, γ =0. Hence

Π5 = bs-1= b/s, our “squareness” parameter Π1 again!


Table for finding Π6

Quantities

sα

MEPβ

ωγ
P

Dimensions

[L]

[ML-1T-2]

[T-1]

[ML2T-3]

M

0

β

0

1

L

α

β

0

2

T

0

-2β

γ

-3

This gives α = -3, β = -1, γ = -1. Hence

Π6 = Ps-3MEP-1ω-1 = P/ωs3MEP

And finally, Π7 is the number of cylinders, N, again, just as in Π2;.


Specific Weight Analysis


What have we learned? Well, assuming that our initial catalog of relevant quantities was correct (and keeping in mind that if we erred in choosing those quantities, all our subsequent work will be in error), we now have grounds for suspecting that the “squareness” of the engine and the number of cylinders that it has are important parameters influencing the power/weight ratio, as these Πs show up in both the weight and the power analyses. That does not mean that we can ignore the others, of course, because if our initial choice of quantities was correct all the Πs are meaningful. Fortunately, though, because of the duplication of two Πs, the total number of independent nondimensional groups is only five instead of the seven that analysis led us to expect. We started with ten quantities to describe our system, and ended up with half that number—a good result for an hour’s work.

But what Π shall we use to represent our dependent parameter, weight per
unit mass? Well, we have a
Π for the engine mass and another for engine
power, and we know that products and ratios of nondimensional parameters
are themselves nondimensional, so we can simply define a new
Π as the ratio
of Π3 to Π6:

Π8 = Π36 = (M/ρb3)/(P/ωs3MEP) = (M/P)(ωMEP/ρ)(s/b)3
Now notice that this new Π contains the cube of one of our independent
Πs, namely “squareness.” We eliminate it by dividing Π8 by Π1 cubed, which
we can do because the quotient of two
Πs is still a Π. That leaves

Π8 = (M/P)(ωMEP/ρ)

to represent our dependent parameter. If we’ve done right so far, we can
expect this Π to be some function of the remaining Πs. In mathematical
notation:

Π8 = F(Π1, Π2, Π4), or

(M/P)(ωMEP/ρ) = F(b/s, N, σ/p)

*Buckingham, E.: “Model Experiments and the Forms of Empirical Equations.” ASME Paper 1487, June 1915 (Published in ASME Transactions, Volume 37, 1916). Download the entire Volume 37 from The Internet Archive

See also Kline, Stephen J.: Similitude and Approximation Theory. McGraw-Hill, 1965

and Bridgman, P.W.: Dimensional Analysis. Yale University Press, 1931

December 5, 2009

Muscle Powered Blimps

Filed under: Aeronautics,Lighter than Air — piolenc @ 4:44 pm

An Introduction to

Muscle Powered Ultralight Gas Blimps

by Robert (“Rex”) Rechs

Rex is a long-time member of the Association of Balloon and Airship Constructors (ABAC), a contributor to Aerostation magazine and an experienced LTA builder, rigger and pilot. This is in addition to his lifelong work in every phase of aviation as both pilot and mechanic.

This volume is intended to be a companion to his Building Small Gas Blimps, but can be read alone if you do not plan to build this kind of machine, but only want to learn about it.

CONTENTS:

Section Title Page
Foreword 1
Introduction 2
Acknowledgments 2
Ultralights 3
History 5
Design Considerations 35
Details and Compromises 47
Ergonomics 87
Rocket Science for Pedal Power 87
The White Dwarf man-powered blimp 140
The Brazilian Dirigible Caloi 179
Vintage Designs 186
Shimano Bicycle Parts Catalog 190
Contacts 210
Bibliography 211
Related Books 212
Organizations 213
Conclusion 214
About the Author 215

Softbound – price: $40 (US currency).

TO ORDER

Order on-line

Build Your Own Gas Blimp – really!

Filed under: Aeronautics,Lighter than Air — piolenc @ 4:21 pm

Building Small Gas Blimps

 

by Robert (“Rex”) Rechs

Rex is a long-time member of the Association of Balloon and Airship Constructors (ABAC), a contributor to Aerostation magazine and an experienced LTA builder, rigger and pilot. This is in addition to his lifelong work in every phase of aviation as both pilot and mechanic. There is probably not another individual on the planet who could have put together such a complex and comprehensive book project.  First published in the late 70’s and extensively revised in 1997, this book was published and promoted exclusively by its author until recently, when he kindly offered the publishing rights to ABAC (copyright remains with the author and ABAC’s use is nonexclusive; for permissions, contact the author directly at r.recks@juno.com).  The book includes detailed lists of materials, parts and suppliers. It is extensively illustrated.

Building a blimp – any manned aircraft, for that matter – is a non-trivial undertaking. What is remarkable about this book is that it actually puts the task within the reach of a determined and patient amateur builder.

ABAC’s edition has essentially the same contents as the 1997 revision. Changes made by ABAC are primarily cosmetic: we’ve numbered the pages, added a table of contents and corrected some typographical errors. The new edition is re-published through CreateSpace, an Amazon affiliate. Only soft covers are available.

CONTENTS:

Section Title Page
Foreword 1
Preface 2
Outline. 3
Glossary 4
Blimp History 6
Design Criteria 7
Ultralights 9
Standard Type Certification 10
Standard Sizes 11
Materials 13
Workmanship 16
Ground Support 22
Masts 29
Hangars 43
Gondola/Airframe 48
Instruments 70
Instruments &
Controls—-Suppliers
80
Engines 82
Ducted Fans 86
2-Stroke Engines—-Suppliers 88
4-Stroke Engines—-Suppliers 89
Propellers—-Suppliers 90
Valves and Pressure System 91
Fins 102
Envelopes 125
Fabric & Webbing—-Suppliers 138
Assembly Procedure 145
Bibliography 153
Airship Fin & Rudder Loads
(BuAer LTA Design Memorandum
No. 169)
155
Gas Airship Parts—-Suppliers 160
Catalog Section. 161
Airship Technical Notes. 185

Soft cover – price: $40 (US currency).

TO ORDER

Order on-line

Model Hot-Air Blimps, a cheap intro to LTA

Filed under: Aeronautics,Lighter than Air — piolenc @ 2:43 pm

Model Hot-Air Blimps: How to Build and Fly Them

by Don James

The ABAC gets many inquiries about plans, kits and books for building model hot-air balloons and airships; some even want to build gas free-flight or RC balloon or airship models. On the gas blimp side of things, we don’t hesitate to recommend the excellent Peck Polymers products (nope, we’re not stockholders!), but for our mostly young inquirers on limited budgets we have not had much to offer.

Enter Don James, founding member of ABAC, graphic artist and perennial LTA enthusiast. In 1980 he and ABAC’s then-President George Wright prepared a 20-page booklet for publication by ABAC. Urgent personal business forced the book to be shelved. The master was misfiled in a folder marked “Promo Pamphlets, etc.” and that was that until Yours Truly, trying to put order in the files, stumbled across it.

Don had set out to provide his own children with a cheap and safe pastime that would teach skills and provide instruction, along with hours of fun. The result was an extensive series of free-flight hot-air blimps made of cheap, commonly-available materials and requiring only ordinary tools for construction. The manual that he prepared based on his experiments is extensively illustrated and provides step-by-step instructions not only for building, but also for designing model blimps. For those who don’t want to design their first project, plans for Don’s “Alpha Blimp” are provided in a tabloid-size centerspread. Instructions for building the “firepot” or heat source, which burns old newspaper, are also included. Inflation and flying directions are copiously illustrated, and the final section diagrams modifications for adding a small model-airplane engine for powered flight.

The book has been re-typeset using the desktop publishing technology that was more or less a dream in 1980. Don’s excellent line drawings and well-exposed b&w photos of his own ships have come through in fine shape and have simply been scanned and incorporated as is. Thirty pages, 6″ x 9″ softcover book, dwgs, photos. US $14 postpaid.

Ordering

  • Order on-line, through Amazon/CreateSpace, using your credit card.

Can You Really Build a Hot-Air Balloon?

Filed under: Aeronautics,Lighter than Air — piolenc @ 2:12 pm

The answer is a qualified “yes.” Here’s what the rigger who literally wrote the book on this subject has to say in the Foreword to his three-and-a-half-volume compendium Build Your Own Balloon:

“This book has been prepared as a guide for the aspiring balloon owner who does not realize the many considerations and details that go into its construction. This book has not been written to encourage individuals to undertake such a  project, but by explaining engineering details it is hoped that you will consult a commercial manufacturer of FAA Type Certificated sport balloons for quality made equipment.

It is realized that many readers will give serious consideration to homebuilt construction; which is, of course, a noble undertaking. Not however without the pitfalls of high cost and poor balloon life; or worse yet, unsafe equipment that the FAA may not let you fly, or may subject to severe operating limitations.

So yes, it can be done, but not just anybody should do it. The author quoted above makes it clear that most of the people who read his book should probably NOT undertake the construction of a balloon. Traits required to do the job successfully are persistence, patience and above all the ability to critically evaluate one’s own work, or at least to listen with an open mind to constructive criticism by others.

Few things look simpler than a hot-air balloon. There’s this big bag filled with hot air, a basket underneath, something to provide heat (where did I put that old camp stove…?). No problem, right? All that’s left to procure is the sandwiches and Champagne.

In fact, a thermal balloon draws on a wide range of skills, from harness-work to welding, and an equally wide range of knowledge from trigonometry to aerostatics to navigation. No matter how much you already know, there will be skills to be acquired and tasks to be contracted out, and you had better be prepared to ruthlessly throw away your own first efforts and start over, because your life will depend on your unwillingness to compromise quality.

Yet even the majority who take things no further than buying the book will learn much that is helpful about the sport of thermal ballooning and about the critical design points of balloons and their support equipment. Balloons are not cheap – that’s the motive for considering building one yourself – and the ability to critically evaluate somebody else’s work could save hundreds, perhaps thousands of dollars, and maybe a life or two.

“Balloon Books”

Build Your Own Balloon Vols I-III + Appendices

Copyright 1996 A.B.A.C.

Compiled by a member of the Association of Balloon and Airship Constructors (ABAC), a licensed balloon and airship pilot, rigger and constructor, these books assemble in one three-volume set the information essential to any amateur considering the construction of a hot-air balloon. They include detailed lists of materials, parts and suppliers. The Appendices contain a brief discussion of gas balloons.

CONTENTS:

Volume I – Design Criteria contains engineering and safety data. Includes international standards, a review of the state of the art and many time-saving ideas and techniques. (180 pages)

Volume II – Materials & Suppliers contains a comprehensive list of components, specifications and prices [note: prices will be out-of-date], plus information on how and where to order. (206 pages)

Volume III – Plans & Construction contains detailed production drawings, assembly instructions and pictures of most components. (175 pages)

Appendix I – Notes on ultralight hot-air balloon construction.

Appendix II – Notes on spherical gas balloon construction.
[Appendices are bound together in one thin volume, 64 pages]

A word of warning: these books are crudely produced. They are sold for their information content, not their beauty!

TO ORDER

Order on-lineOrder on-line through Amazon/CreateSpace

Volume 1 ($30): https://www.createspace.com/5854671

Volume 2 ($30): https://www.createspace.com/5875635

Volume 3 ($30): https://www.createspace.com/6393966

Appendices ($20): https://www.createspace.com/6396032

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